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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmap0 | Structured version Visualization version GIF version |
Description: Value of the projective map of a Hilbert lattice at lattice zero. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.) |
Ref | Expression |
---|---|
pmap0.z | ⊢ 0 = (0.‘𝐾) |
pmap0.m | ⊢ 𝑀 = (pmap‘𝐾) |
Ref | Expression |
---|---|
pmap0 | ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | pmap0.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
3 | 1, 2 | atl0cl 34590 | . . 3 ⊢ (𝐾 ∈ AtLat → 0 ∈ (Base‘𝐾)) |
4 | eqid 2622 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
5 | eqid 2622 | . . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
6 | pmap0.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
7 | 1, 4, 5, 6 | pmapval 35043 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 0 ∈ (Base‘𝐾)) → (𝑀‘ 0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 }) |
8 | 3, 7 | mpdan 702 | . 2 ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 }) |
9 | 4, 2, 5 | atnle0 34596 | . . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑎 ∈ (Atoms‘𝐾)) → ¬ 𝑎(le‘𝐾) 0 ) |
10 | 9 | nrexdv 3001 | . . . 4 ⊢ (𝐾 ∈ AtLat → ¬ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 ) |
11 | rabn0 3958 | . . . 4 ⊢ ({𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ ∃𝑎 ∈ (Atoms‘𝐾)𝑎(le‘𝐾) 0 ) | |
12 | 10, 11 | sylnibr 319 | . . 3 ⊢ (𝐾 ∈ AtLat → ¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅) |
13 | nne 2798 | . . 3 ⊢ (¬ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } ≠ ∅ ↔ {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅) | |
14 | 12, 13 | sylib 208 | . 2 ⊢ (𝐾 ∈ AtLat → {𝑎 ∈ (Atoms‘𝐾) ∣ 𝑎(le‘𝐾) 0 } = ∅) |
15 | 8, 14 | eqtrd 2656 | 1 ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 {crab 2916 ∅c0 3915 class class class wbr 4653 ‘cfv 5888 Basecbs 15857 lecple 15948 0.cp0 17037 Atomscatm 34550 AtLatcal 34551 pmapcpmap 34783 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-preset 16928 df-poset 16946 df-plt 16958 df-glb 16975 df-p0 17039 df-lat 17046 df-covers 34553 df-ats 34554 df-atl 34585 df-pmap 34790 |
This theorem is referenced by: pmapeq0 35052 pmapjat1 35139 pol1N 35196 pnonsingN 35219 |
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